Optimal. Leaf size=89 \[ -\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}+\frac {(\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {652, 632, 212}
\begin {gather*} \frac {(\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 652
Rubi steps
\begin {align*} \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}-\frac {(\text {b1} c-b \text {c1}) \int \frac {1}{a+2 b x+c x^2} \, dx}{2 \left (b^2-a c\right )}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}+\frac {(\text {b1} c-b \text {c1}) \text {Subst}\left (\int \frac {1}{4 \left (b^2-a c\right )-x^2} \, dx,x,2 b+2 c x\right )}{b^2-a c}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}+\frac {(\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 88, normalized size = 0.99 \begin {gather*} \frac {\frac {-b \text {b1}+a \text {c1}-\text {b1} c x+b \text {c1} x}{a+x (2 b+c x)}+\frac {(-\text {b1} c+b \text {c1}) \tan ^{-1}\left (\frac {b+c x}{\sqrt {-b^2+a c}}\right )}{\sqrt {-b^2+a c}}}{2 \left (b^2-a c\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 103, normalized size = 1.16
method | result | size |
default | \(\frac {\left (-2 b \mathit {c1} +2 \mathit {b1} c \right ) x +2 b \mathit {b1} -2 a \mathit {c1}}{\left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )}+\frac {\left (-2 b \mathit {c1} +2 \mathit {b1} c \right ) \arctan \left (\frac {2 c x +2 b}{2 \sqrt {a c -b^{2}}}\right )}{\left (4 a c -4 b^{2}\right ) \sqrt {a c -b^{2}}}\) | \(103\) |
risch | \(\frac {-\frac {\left (b \mathit {c1} -\mathit {b1} c \right ) x}{2 \left (a c -b^{2}\right )}-\frac {a \mathit {c1} -b \mathit {b1}}{2 \left (a c -b^{2}\right )}}{c \,x^{2}+2 b x +a}+\frac {\ln \left (\left (-c^{2} a +b^{2} c \right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}-a b c +b^{3}\right ) b \mathit {c1}}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (-c^{2} a +b^{2} c \right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}-a b c +b^{3}\right ) \mathit {b1} c}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (c^{2} a -b^{2} c \right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}+a b c -b^{3}\right ) b \mathit {c1}}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left (\left (c^{2} a -b^{2} c \right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}+a b c -b^{3}\right ) \mathit {b1} c}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}\) | \(262\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 210 vs.
\(2 (81) = 162\).
time = 0.44, size = 447, normalized size = 5.02 \begin {gather*} \left [-\frac {2 \, b^{3} b_{1} - 2 \, a b b_{1} c - {\left (a b_{1} c - a b c_{1} + {\left (b_{1} c^{2} - b c c_{1}\right )} x^{2} + 2 \, {\left (b b_{1} c - b^{2} c_{1}\right )} x\right )} \sqrt {b^{2} - a c} \log \left (\frac {c^{2} x^{2} + 2 \, b c x + 2 \, b^{2} - a c + 2 \, \sqrt {b^{2} - a c} {\left (c x + b\right )}}{c x^{2} + 2 \, b x + a}\right ) - 2 \, {\left (a b^{2} - a^{2} c\right )} c_{1} + 2 \, {\left (b^{2} b_{1} c - a b_{1} c^{2} - {\left (b^{3} - a b c\right )} c_{1}\right )} x}{4 \, {\left (a b^{4} - 2 \, a^{2} b^{2} c + a^{3} c^{2} + {\left (b^{4} c - 2 \, a b^{2} c^{2} + a^{2} c^{3}\right )} x^{2} + 2 \, {\left (b^{5} - 2 \, a b^{3} c + a^{2} b c^{2}\right )} x\right )}}, -\frac {b^{3} b_{1} - a b b_{1} c - {\left (a b_{1} c - a b c_{1} + {\left (b_{1} c^{2} - b c c_{1}\right )} x^{2} + 2 \, {\left (b b_{1} c - b^{2} c_{1}\right )} x\right )} \sqrt {-b^{2} + a c} \arctan \left (-\frac {\sqrt {-b^{2} + a c} {\left (c x + b\right )}}{b^{2} - a c}\right ) - {\left (a b^{2} - a^{2} c\right )} c_{1} + {\left (b^{2} b_{1} c - a b_{1} c^{2} - {\left (b^{3} - a b c\right )} c_{1}\right )} x}{2 \, {\left (a b^{4} - 2 \, a^{2} b^{2} c + a^{3} c^{2} + {\left (b^{4} c - 2 \, a b^{2} c^{2} + a^{2} c^{3}\right )} x^{2} + 2 \, {\left (b^{5} - 2 \, a b^{3} c + a^{2} b c^{2}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 323 vs.
\(2 (75) = 150\).
time = 0.53, size = 323, normalized size = 3.63 \begin {gather*} \frac {\sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) \log {\left (x + \frac {- a^{2} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) + 2 a b^{2} c \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) - b^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) + b^{2} c_{1} - b b_{1} c}{b c c_{1} - b_{1} c^{2}} \right )}}{4} - \frac {\sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) \log {\left (x + \frac {a^{2} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) - 2 a b^{2} c \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) + b^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) + b^{2} c_{1} - b b_{1} c}{b c c_{1} - b_{1} c^{2}} \right )}}{4} + \frac {- a c_{1} + b b_{1} + x \left (- b c_{1} + b_{1} c\right )}{2 a^{2} c - 2 a b^{2} + x^{2} \cdot \left (2 a c^{2} - 2 b^{2} c\right ) + x \left (4 a b c - 4 b^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 92, normalized size = 1.03 \begin {gather*} -\frac {{\left (b_{1} c - b c_{1}\right )} \arctan \left (\frac {c x + b}{\sqrt {-b^{2} + a c}}\right )}{2 \, {\left (b^{2} - a c\right )} \sqrt {-b^{2} + a c}} - \frac {b_{1} c x - b c_{1} x + b b_{1} - a c_{1}}{2 \, {\left (c x^{2} + 2 \, b x + a\right )} {\left (b^{2} - a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 159, normalized size = 1.79 \begin {gather*} \frac {\mathrm {atan}\left (\frac {2\,\left (\frac {\left (4\,b^3-4\,a\,b\,c\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,{\left (a\,c-b^2\right )}^{5/2}}-\frac {c\,x\,\left (b\,c_{1}-b_{1}\,c\right )}{2\,{\left (a\,c-b^2\right )}^{3/2}}\right )\,\left (a\,c-b^2\right )}{b\,c_{1}-b_{1}\,c}\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{2\,{\left (a\,c-b^2\right )}^{3/2}}-\frac {\frac {a\,c_{1}-b\,b_{1}}{2\,\left (a\,c-b^2\right )}+\frac {x\,\left (b\,c_{1}-b_{1}\,c\right )}{2\,\left (a\,c-b^2\right )}}{c\,x^2+2\,b\,x+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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